Characteristic numbers and the Bott residue

According to the celebrated Hopf index theorem, the Euler characteristic of a smooth manifold is equal to the number of zeros of a vector field on the manifold, each counted with its index. In [41] and [43], Bott generalized the Hopf index theorem to other characteristic numbers such as the Pontryagin numbers of a real manifold and the Chern numbers of a complex manifold.

We will describe Bott's formula only for Chern numbers. Let $M$ be a compact complex manifold of dimension $n$ , and $c_1(M), \dots, c_n(M)$ the Chern classes of the tangent bundle of $M$ . The Chern numbers of $M$ are the integrals $\int_M \phi (c_1(M), \dots , c_n(M))$ , as $\phi$ ranges over all weighted homogeneous polynomials of degree $n$ . Like the Hopf index theorem, Bott's formula computes a Chern number in terms of the zeros of a vector field $X$ on $M$ , but the vector field must be holomorphic and the counting of the zeros is a little more subtle.

For any vector field $Y$ and any $C^{\infty}$ function $f$ on $M$ , the Lie derivative $\mathcal{L}_{X}$ satisfies:

$$\mathcal{L}_{X}(fY)= (Xf)Y+f\mathcal{L}_{X}Y.$$

It follows that at a zero $p$ of $X$ ,

$$(\mathcal{L}_{X}fY)_p = f(p) (\mathcal{L}_{X}Y)p.$$

Thus, at $p$ , the Lie derivative $\mathcal{L}_{X}$ induces an endomorphism

$$L_p : T_pM \to T_p M$$

of the tangent space of $M$ at $p$ . The zero $p$ is said to be nondegenerate if $L_p$ is nonsingular.

For any endomorphism $A$ of a vector space $V$ , we define the numbers $c_i (A)$ to be the coefficients of its characteristic polynomial:

$$\det (I+tA) = \sum c_i (A) t^i.$$

Bott's Chern number formula is as follows. Let $M$ be a compact complex manifold of complex dimension $n$ and $X$ a holomorphic vector field having only isolated nondegenerate zeros on $M$ . For any weighted homogeneous polynomial $\phi (x_1, \dots, x_n)$ , $\deg x_i = 2i$ ,

$$\int_M \phi (c_1(M), \dots, c_n(M)) = \sum_p \dfrac{\phi (c_1 (L_p), \dots, c_n(L_p))}{c_n(L_p)},$$ (2)

summed over all the zeros of the vector field. Note that by the definition of a nondegenerate zero, $c_n(L_p)$ , which is $\det L_p$ , is nonzero.

In Bott's formula, if the polynomial $\phi$ does not have degree $2n$ , then the left-hand side of (2) is zero, and the formula gives an identity among the numbers $c_i(L_p)$ . For the polynomial $\phi (x_1, \dots, x_n) = x_n$ , Bott's formula recovers the Hopf index theorem:

$$\int_M c_n (M) = \sum_p \dfrac{c_n(L_p)}{c_n(L_p)} = \char93$$    zeros of $$X.$$

Bott's formula (2) is reminiscent of Cauchy's residue formula and so the right-hand side of (2) may be viewed as a residue of $\phi$ at $p$ .

In [43] Bott generalized his Chern number formula (2), which assumes isolated zeros, to holomorphic vector fields with higher-dimensional zero sets and to bundles other than the tangent bundle (a vector field is a section of the tangent bundle).